The n-dimensional Laplace Equation

The n-dimensional Laplace equation is of the form:

$$\nabla^{2}_{}$$$$\phi$$ = $$\phi_{x_1 x_1}^{}$$ + $$\phi_{x_2 x_2}^{}$$ + ... + $$\phi_{x_n x_n}^{}$$ = 0

The determining systems for this equation are the simplest that we have posted Lie symmetry determining systems for.

The determining system consists of the dependent variables $\Phi$ and X01, X02,..., Xn which depend on the independent variables $\phi$ and x01, x02,..., xn.

The systems for n = 3..16 can be downloaded here (the files in the archive are named laplace_n). It should be emphasized that these are test systems only, as they are homogeneous constant coefficient PDE in the infinitesimals, and are relatively easy to bring to simplified form (as can be seen by the run-times below).

	Input		PureElim
Dimension	Equations	Length	Time (sec)	Mem (MB)
3	34	2723	0.01	0.70
4	58	5027	0.02	0.72
5	88	8211	0.04	0.75
6	124	12395	0.08	0.78
7	166	17699	0.15	0.86
8	214	24243	0.23	0.89
9	268	32147	0.39	1.00
10	328	41531	0.70	1.24
11	394	52515	1.10	1.39
12	466	65219	1.73	1.59
13	544	79763	3.19	1.88
14	628	96267	5.09	2.72
15	718	114851	8.77	3.43
16	814	135635	14.22	3.92